Non-Bloch band theory and bulk-edge correspondence in non-Hermitian systems
aa r X i v : . [ c ond - m a t . m e s - h a ll ] N ov Prog. Theor. Exp. Phys. , 00000 (12 pages)DOI: 10.1093 / ptep/0000000000 Non-Bloch band theory and bulk-edgecorrespondence in non-Hermitian systems
Kazuki Yokomizo and Shuichi Murakami Department of Physics, Tokyo Institute of Technology, 2-12-1 Ookayama,Meguro-ku, Tokyo, 152-8551, Japan ∗ E-mail: [email protected] TIES, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo,152-8551, Japan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
In this paper, we review our non-Bloch band theory in one-dimensional non-Hermitiantight-binding systems. In our theory, it is shown that in non-Hermitian systems, theBrillouin zone is determined so as to reproduce continuum energy bands in a large openchain. By using simple models, we explain the concept of the non-Bloch band theoryand the method to calculate the Brillouin zone. In particular, for the non-HermitianSu-Schrieffer-Heeger model, the bulk-edge correspondence can be established betweenthe topological invariant defined from our theory and existence of the topological edgestates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Subject Index xxxx, xxx
1. Introduction
In recent years, interest in studies of non-Hermitian systems has been rapidly growing bothin theories and in experiments. The non-Hermitian Hamiltonian is useful for studying non-equilibrium systems and open systems, which exchange energies and particles with externalenvironment. Non-Hermitian systems emerge in various fields of classical physics and quan-tum physics [1]. In classical systems, gain and loss lead to non-Hermitian terms in eigenvalueequations. On the other hand, in quantum systems, one of the origins of non-Hermiticityis many-body correlation effect. For example, in strongly correlated electron systems, wecan get non-Hermitian Hamiltonian by incorporating the imaginary part of the self-energyrepresenting the lifetime of quasiparticle into one-body Hamiltonian [2–4]. One of the mostintriguing topics is how non-Hermitian effects affect topological physics. For example, someprevious works proposed new definitions of a gap in non-Hermitian systems, and in termsof this definition, topological classifications of gapped phases and gapless phases are givenunder some symmetries [4–11]. Furthermore, new topological invariants can appear thanksto unique features of non-Hermitian systems such as non-Hermitian degeneracies [12].Among theoretical works of non-Hermitian topological systems, in particular, violation ofthe bulk-edge correspondence has been a long-standing issue in this field, and the reasonsfor this violation have been under debate [13–19]. One of the controversies is that in mostof the previous works, the Bloch wave number k has been treated as real in non-Hermitiansystems, similarly to Hermitian ones. In contrast to these previous works, it was proposed © The Author(s) 2012. Published by Oxford University Press on behalf of the Physical Society of Japan.This is an Open Access article distributed under the terms of the Creative Commons Attribution License(http://creativecommons.org/licenses/by-nc/3.0), which permits unrestricted use,distribution, and reproduction in any medium, provided the original work is properly cited. hat k takes complex values in non-Hermitian systems in order to describe electronic statesin a long open chain because the energy spectrum in a periodic chain and that in an openchain are different [20, 21]. Then the value of β ≡ e ik is confined on a loop on the complexplane so as to reproduce continuum energy bands in a large open chain. This loop is ageneralization of the conventional Brillouin zone and is called generalized Brillouin zone(GBZ). Furthermore the eigenstates of the non-Hermitian Hamiltonian do not necessarilyextend over the whole system but are localized at either end of the chain. This phenomenonis called the non-Hermitian skin effect [20]. After the proposal of this effect, intriguingproperties of the non-Hermitian skin effect have been studied intensively both in theoriesand experiments [22–29].In our work [21], we established a non-Bloch band theory in a one-dimensional (1D) non-Hermitian tight-binding system. We showed how to determine the GBZ C β for β ≡ e ik , k ∈ C . In the present paper, we review this non-Bloch band theory with simple examples. Inorder to understand the concept of our theory, we explain in detail how C β can be obtainedin simple models. Furthermore we can establish the bulk-edge correspondence in the non-Hermitian Su-Schrieffer-Heeger (SSH) model from the topological invariant defined by C β .
2. Simple model (a)(b) ・・・・・・ t R t L ・・・・・・ t R t L (c-1) ββ ( m ) E ( m ) E L → ∞ (c-2) L ・・・ Im ImRe Re
Fig. 1 (a) Simple model in an infinite open chain. (b) Simple model in a finite open chainwith the system size L . (c-1) Schematic figures of the distribution of β ( m ) and that of E ( m ) with m = 1 , · · · , L . In the limit of L → ∞ , β ( m ) forms a circle with the radius p t R /t L , and E ( m ) forms the continuum energy band with the range of (cid:2) − √ t R t L , √ t R t L (cid:3) , shown in(c-2).In this section, we study the constructions of the GBZ and of the continuum energy bandsin a simple non-Hermitian tight-binding model. This model is known as the Hatano-Nelsonmodel without disorder [30]. The real-space Hamiltonian of this system is given by H = L − X n =1 (cid:16) t R c † n +1 c n + t L c † n c n +1 (cid:17) , (1)where t R , t L ∈ R are nearest-neighbor asymmetric hopping amplitudes to the right and tothe left, respectively (Fig. 1(a)). We show the schematic figure of this system in a finite openchain with the system size L in Fig. 1(b). The real-space eigen-equation H | ψ i = E | ψ i , forthe eigenvector | ψ i = ( ψ , · · · , ψ L ) T , can be written as t R ψ n − + t L ψ n +1 = Eψ n , ( n = 1 , · · · , L ) , ψ = ψ L +1 = 0 . (2) rom the theory of linear difference equations, a general solution of the recursion equation(2) is written as ψ n = ( β ) n φ (1) + ( β ) n φ (2) , (3)where β j ( j = 1 ,
2) are the solutions of the equation t R β − + t L β = E. (4)Together with the open boundary conditions ψ = 0 and ψ L +1 = 0, we can get β ) L +1 ( β ) L +1 ! φ (1) φ (2) ! = ! (5)and obtain a boundary equation, which represents the boundary conditions at the two ends,as ( β /β ) L +1 = 1 so that the coefficients φ (1) , φ (2) take nonzero values. Then one can get β β = e iθ m , (cid:18) θ m = mπL + 1 , m = 1 , · · · , L (cid:19) . (6)Therefore, from Eq. (4), β j ( j = 1 ,
2) can be written as β ( m )1 = r e iθ m , β ( m )2 = r e − iθ m , (7)where r = | β , | = p t R /t L because β β = t R /t L from the Vieta’s formula, and the eigen-state (3) and the eigenenergy (4) are written as ψ ( m ) n ∝ (cid:16) r e iθ m (cid:17) n − (cid:16) r e − iθ m (cid:17) n ∝ r n sin nθ m , E ( m ) = 2 √ t R t L cos θ m , (8)respectively. From Eq. (8), the distribution of the discrete eigenstates and that of the discreteenergy levels are shown in Fig. 1(c-1).Now, as the system size L becomes larger, these eigenstates and energy levels becomedense. Finally, in the limit of L → ∞ (Fig. 1(a)), the energy levels form a continuum energyband as shown in Fig. 1(c-2), leading to the form of the continuum eigenstates as β = r t R t L e iθ , β = r t R t L e − iθ (9)from Eq. (7), and to that of the continuum energy band as E = 2 √ t R t L cos θ (10)from Eq. (4) by changing the parameter θ ∈ R .By a comparison with a Hermitian case, which is realized when t R = t L , we can intuitivelyunderstand these results in the viewpoint of the Bloch band theory. Equation (3) meansthat β (= β , β ) can be related with the Bloch wave number k by β = e ik . In this sense,the distribution of β (= e ik ), which is called generalized Brillouin zone (GBZ), gives a non-Hermitian extension of the Brillouin zone. In 1D Hermitian systems, the wave number is real,and the GBZ is always a unit circle. On the other hand, in the present case, the Brillouin zoneis not a unit circle, meaning that the corresponding wave number is not real. We note thatthe eigenstate for the finite chain of the simple model is a superposition of two “plane waves”with β and β , and these two values satisfy Eq. (4) with the same energy E . Therefore,once the GBZ is shown, the energy eigenvalues E are calculated from Eq. (4). As a result,the continuum energy band is formed in the range of (cid:2) − √ t R t L , √ t R t L (cid:3) . ere we comment the dependence of the above results on boundary conditions. If we changethe boundary conditions, ψ = a and ψ L +1 = b as an example, the form of the boundaryequation in a finite open chain is modified. Then the energy levels (8) are also modified,which means that the energy levels in a finite open chain depend on boundary conditions.Nevertheless, in the limit of L → ∞ , the continuum energy bands and the GBZ are inde-pendent of boundary conditions in an open chain. Therefore they can be obtained as shownin Fig. 1(c-2) under any boundary conditions in an open chain.
3. Non-Bloch band theory ・・・ ・・・ q ・・・ ・・・・・・ ・・・ q ・・・ ・・・ q + N + N ・・・ ・・・ L Finite open chain bÎ C b Infinite open chain E, (b-1) (b-2) Continuumenergy bands E ImRe(a)
Fig. 2 (a) One-dimensional tight-binding system. A unit cell includes q degree of freedom,and the range of hopping is N . (b) Schematic figures of (b-1) energy levels in a finite openchain with various system sizes L and of (b-2) continuum energy bands in an infinite openchain. The vertical axis represents the distribution of the complex energy E .We can generalize the result in the previous section to general 1D non-Hermitian systems.We start with a 1D tight-binding system with spatial periodicity as shown in Fig. 2(a). Aunit cell is composed of q degrees of freedom, such as sublattices, spins, or orbitals, and theelectrons hop to the N -th nearest unit cells. Then its Hamiltonian can be written as H = X n N X i = − N q X µ,ν =1 t i,µν c † n + i,µ c n,ν , (11)where c † n,µ ( c n,µ ) is a creation (an annihilation) operator of an electron with an index µ ( µ =1 , · · · , q ) in the n -th unit cell, and t i,µν is a hopping amplitude to the i -th nearest unitcell. This Hamiltonian can be non-Hermitian, meaning that t i,µν is not necessarily equal to t ∗− i,νµ . In this situation, the real-space eigen-equation is written as H | ψ i = E | ψ i , where theeigenvector is given by | ψ i = ( · · · , ψ , , · · · , ψ ,q , ψ , , · · · , ψ ,q , · · · ) T . Then, as is similar tothe model in the previous section, | ψ i can be represented as a linear combination: ψ n,µ = X j φ ( j ) n,µ , φ ( j ) n,µ = ( β j ) n φ ( j ) µ , ( µ = 1 , · · · , q ) , (12)where β = β j is a solution of the characteristic equation defined asdet [ H ( β ) − E ] = 0 , [ H ( β )] µν = N X i = − N t i,µν ( β ) i , ( µ, ν = 1 , · · · , q ) , (13) nd the eigen-equation of the matrix H ( β ) can be explicitly rewritten as q X ν =1 [ H ( β )] µν φ ν = Eφ ν , ( µ = 1 , · · · , q ) . (14)We note that H ( β ) becomes the Bloch Hamiltonian if we rewrite it in terms of the conven-tional Bloch wave number k . Furthermore the equation (13) is an algebraic equation for β with an even degree 2 M = 2 qN .Now we explain the concept of the non-Bloch band theory. The resulting energy levelsare discrete in a finite open chain with a system size L as shown in Fig. 2(b-1). Here, as L becomes larger, the energy levels become dense and asymptotically continuous. Finally,in the limit of L → ∞ , the continuum energy bands are formed as shown in Fig. 2(b-2).Then the asymptotic distribution of β for L → ∞ is the GBZ C β . We note that in thiscase, the absolute value of β is not necessarily unity, and C β is obtained as a loop on thecomplex plane. The key question is how to construct the GBZ for the system consideredhere. From the argument so far, we may need to calculate the energy levels for finite L andstudy its asymptotic behavior for L → ∞ . This is a cumbersome procedure, and the resultmay possibly depend on boundary conditions.Remarkably, in Ref. [21], we found a method to calculate the GBZ C β , without goingthrough a calculation on a finite open chain with the system size L . This largely simplifiesthe calculation. It is worth noting that C β is independent of boundary conditions. Thus,while energy levels in a finite open chain depend on boundary conditions, their asymptoticbehaviors do not.Below we explain a way to calculate the GBZ C β , which determines continuum energybands. Let β j ( j = 1 , · · · , M ) be the solutions of the equation (13). When we number the2 M solutions so as to satisfy | β | ≤ | β | ≤ · · · ≤ | β M − | ≤ | β M | , (15)we find that the condition for continuum energy bands is given by | β M | = | β M +1 | , (16)and the trajectories of β M and β M +1 give C β . The example in Sec. 2 is a special case with M = 1. In Secs. 3.2 and 3.3, we will show some examples of C β and the continuum energybands calculated by using the condition (16). Here we note that although the eigenener-gies for the continuum energy bands are obtained from Eq. (13) by putting β = β M and β = β M +1 , the eigenvectors of Eq. (14) are not eigenstates of the Hamiltonian (11). Instead,the eigenstates of the Hamiltonian (11) is given by Eq. (12), which involves the termswith β = β , · · · , β M . Nonetheless, the non-Bloch band theory explained here says thatthe eigenenergies for the continuum energy bands are determined by β M and β M +1 , andthat the GBZ C β and a set of the eigenenergies are independent of boundary conditions inan open chain. Thus, in the calculation of C β , we do not need to solve the eigenvalue prob-lem in Eq. (14). The matrix H ( β ) is introduced here in order to express the characteristicequation (13).While the conclusion in Eq. (16) is simple, its derivation presented in the SupplementalMaterial of Ref. [21] is lengthy. Therefore, instead of reproducing it here, we explain itsoutline. First, we impose the given boundary conditions in an open chain onto the eigenvector ψ i with Eq. (12) to get a set of the linear equations for φ ( j ) µ . Then the condition for thisset of the linear equations to have a nontrivial solution yields an equation for β j ’s, calledboundary equation. This boundary equation is a complicated equation dependent on theboundary conditions, giving discrete energy levels. Nonetheless, in the limit of a large systemsize, L → ∞ , we expect that the energy levels become dense and eventually form continuumenergy bands. Therefore we impose a condition that the solutions of the boundary equationshould have an asymptotically dense set. Then we get the Eq. (16) which is eventuallyindependent of the form of open boundary conditions. More details are presented in theSupplemental Material of Ref. [21].The condition for continuum energy bands (16) can be regarded as a condition for for-mation of a standing wave. Equation (16) means that the decay lengths of the eigenstatescorresponding to β M and β M +1 are equal, so that the wave function vanishes at both endsof an open chain. For example, in the model in Sec. 2, the wave function (8) represents astanding wave apart from the factor r n , as a superposition of two counterpropagating “planewaves”. Furthermore the condition (16) is physically reasonable in several aspects. Firstly,this condition does not depend on any boundary conditions in an open chain. Secondly, inthe Hermitian limit, we can rewrite Eq. (16) to the well known result, i.e. | β M | = | β M +1 | = 1.For example, in the model in Sec. 2 with the case of t R = t L , the model becomes Hermitian,and the GBZ becomes a unit circle, identified with the conventional Brillouin zone.Finally we mention the case that the characteristic equation (13) is a reducible algebraicequation. Namely it can be factorized as det [ H ( β ) − E ] = f ( β, E ) · · · f m ( β, E ), where f i ( β, E ) ( i = 1 , · · · , m ) are algebraic equations for β and E . For simplicity, we assumethat they are algebraic equations for β with an even degree 2 M i . In this case, the contin-uum energy bands and the GBZs can be obtained from the conditions | β M i | = | β M i +1 | ( i =1 , · · · , m ) instead of Eq. (16) [31]. Fig. 3
Non-Hermitian Su-Schrieffer-Heeger (SSH) model in an infinite open chain. Thedotted boxes indicate the unit cell. (b)-(d) Generalized Brillouin zone in the non-HermitianSSH model. The values of the parameters are (b) t = 1 . , t = 1 , t = 1 / , γ = 4 / γ = 0; (c) t = 0 . , t = 1 . , t = 1 / , γ = 0, and γ = − /
3; and (d) t = − . , t =0 . , t = 1 / , γ = 5 /
3, and γ = 1 / s written as H = X n h t c † n, A c n +1 , B + (cid:16) t + γ (cid:17) c † n, A c n, B + (cid:16) t − γ (cid:17) c † n +1 , A c n, B + (cid:16) t + γ (cid:17) c † n, B c n +1 , A + (cid:16) t − γ (cid:17) c † n, B c n, A + t c † n +1 , B c n, A i . (17)Henceforth we set all the parameters to be real. For the real-space eigenvector | ψ i =( · · · , ψ , A , ψ , B , ψ , A , ψ , B , · · · ), we can explicitly write the real-space eigen-equation as t ψ n +1 , B + (cid:16) t + γ (cid:17) ψ n, B + (cid:16) t − γ (cid:17) ψ n − , B = Eψ n, A , (cid:16) t + γ (cid:17) ψ n +1 , A + (cid:16) t − γ (cid:17) ψ n, A + t ψ n − , A = Eψ n, B . (18)Here we can take a general ansatz for the wave function as a linear combination: ψ n, A ψ n, B ! = X j ( β j ) n φ ( j )A φ ( j )B ! , (19)where β = β j are the solutions of the characteristic equation det [ H ( β ) − E ] = 0 for thematrix H ( β ) = t β + (cid:16) t + γ (cid:17) + (cid:16) t − γ (cid:17) β − (cid:16) t + γ (cid:17) β + (cid:16) t − γ (cid:17) + t β − . (20)In this case, the characteristic equation h t β + (cid:16) t + γ (cid:17) + (cid:16) t − γ (cid:17) β − i h(cid:16) t + γ (cid:17) β + (cid:16) t − γ (cid:17) + t β − i − E = 0 (21)is a quartic equation for β , having four solutions β i ( i = 1 , · · · ,
4) satisfying | β | ≤ | β | ≤| β | ≤ | β | . In this case of M = 2, the condition for continuum energy bands is given by | β | = | β | . (22)It is obtained by imposing that Eq. (19) satisfying open boundary conditions forms a denseset of solutions at L → ∞ . As emphasized earlier, Eq. (22) does not depend on boundaryconditions in an open chain as shown in the Supplemental Material of Ref. [21].Here the trajectories of β and β give the GBZ C β as shown in Figs. 3 (b)-(d) with variousvalues of the parameters. It is worth mentioning some features of C β in the following. First, asshown in Fig. 3(d), | β | on C β takes both values more than 1 and values less than 1 [23, 24, 31].Here, | β | > | β | <
1) means that the eigenstate is localized at the right (left) end of thechain, representing the non-Hermitian skin effect. Second, C β is a closed loop encircling theorigin on the complex plane [24, 31]. Finally, C β can have the cusps, corresponding to thecases where three solutions of Eq. (21) share the same absolute value.In order to calculate the GBZ, we need to judge the condition (22) for a givenvalue of E . To explain this, we set the values of the parameters as ( t , t , t , γ , γ ) =(3 / , / , / , / , / E = E = 0 .
140 + 0 . i , we obtainthe four solutions of Eq. (21) as( β , β , β , β ) = (0 . − . i, .
064 + 0 . i, − .
506 + 0 . i, − . − . i ) , ( | β | , | β | , | β | , | β | ) = (0 . , . , . , . . (23)Since these solutions satisfy the condition for continuum energy bands (22), we concludethat E is included in the continuum energy bands. On the other hand, when we substitute = E = 0 .
180 + 1 . i into Eq. (21), we obtain the solutions as( β , β , β , β ) = (0 . − . i, − .
052 + 0 . i, − .
52 + 1 . i, − . − . i ) , ( | β | , | β | , | β | , | β | ) = (0 . , . , . , . . (24)In this case, E is not in the continuum energy bands because | β | 6 = | β | .From Eq. (22), we calculate the continuum energy bands as shown in Fig. 5(c). In fact,we can confirm that this result agrees with the energy levels in a finite open chain as shownin Fig. 5(d). In conclusion, it is shown that Eq. (22) is appropriate for the condition forcontinuum energy bands. In Sec. 4, we investigate the topological edge states appearing ina finite open chain as shown in red in Fig. 5(d). t R t L ・・・・・・ t R t L ‘ ‘ Δ ΔΔU U U (b)(a) (c)
Fig. 4 (a) The simple two-band model (25) in an infinite open chain. The dotted boxesindicate the unit cell. (b) Generalized Brillouin zones and (c) continuum energy bands in thetwo-band model (25). They correspond to each other in the same color. We set the valuesof the parameters as t R = 2 , t L = 1 , t ′ R = 1 , t ′ L = 3, and U = ∆ = − β = e ik , k ∈ R . Here we show the splitting of the GBZ in a two-band non-Hermitianmodel as shown in Fig. 4(a), following Ref. [31]. This system has no symmetries. The real-space Hamiltonian can be written as H = X n (cid:16) t R c † n +1 , A c n, A + ∆ c † n, A c n, B + t L c † n +1 , A c n, A + t ′ R c † n +1 , B c n, B + ∆ c † n, B c n, A + U c † n, B c n, B + t ′ L c † n, B c n +1 , B (cid:17) . (25)By the same procedure in Sec. 3.2, we can get the matrix H ( β ) as H ( β ) = t R β − + t L β ∆∆ t ′ R β − + U + t ′ L β ! . (26)In this case, we can get the GBZs and the continuum bands by applying the condition forcontinuum energy bands (22) to the solutions of the characteristic equation det [ H ( β ) − E ] =0, and the results are shown in Figs. 4(b) and (c). One can see that the GBZs split into twocurves, each of which corresponds to the individual band. e note that under some additional symmetries, some bands necessarily share the sameGBZ. For example, the non-Hermitian SSH model (17) has only one GBZ because it hassublattice symmetry (SLS). Here the SLS is defined as Γ H Γ − = − H for a real-space Hamil-tonian (11), where Γ is a unitary matrix satisfying Γ = +1. Namely the eigenenergy of thissystem appears in pairs, ( E, − E ), both of which come from the same GBZ because of theform of the eigenvalue equation (21).
4. Bulk-edge correspondence
Fig. 5
Bulk-edge correspondence in the non-Hermitian Su-Schrieffer-Heeger model withthe values of the parameters as t = 1 / , γ = 5 /
3, and γ = 1 /
3. (a) Phase diagram on the t - t plane. The blue and the white regions represent a topological insulator phase with thewinding number w being 1 and a normal insulator phase with w = 0, respectively. The orangeregion represents the gapless phase. (b) Trajectories ℓ + (red) and ℓ − (blue) on the R planewith t = 1 and t = 1 .
4. The arrows mean the direction of the change of R ± ( β ) as β goesin a counterclockwise manner along the generalized Brillouin zone (GBZ). (c) Continuumenergy bands from the GBZ. We show the results for them along the black arrow in (a) with t = 1 .
4. (d) Energy levels in a finite open chain with the system size L = 100. The red linerepresents the topological edge states.In this section, we discuss the bulk-edge correspondence in the non-Hermitian SSH modelintroduced in Sec. 3.2. It is shown that the topological invariant defined in terms of the GBZcan precisely predict existence of the topological edge states.First of all, we define a topological invariant in a 1D non-Hermitian system with the SLSin a two-band model. Let us start with the real-space Hamiltonian (11). By the procedureexplained in Sec. 3.1, we can get the matrix H ( β ) in the form of Eq. (13) in the bulk ofa large open chain. Then, if we can put the matrix form of the SLS as diag(1 , − H ( β ) of this system as an off-diagonal form; H ( β ) = R + ( β ) R − ( β ) 0 ! , (27)where R ± ( β ) are polynomials of β and β − . The complex wave number can be determined as β ≡ e ik , k ∈ C on the GBZ C β given from Eq.(16). We note that the energy eigenvalues canbe explicitly written as E ± ( β ) = ± p R + ( β ) R − ( β ). Then the topological invariant calledwinding number w is defined as w = − π [arg R + ( β ) − arg R − ( β )] C β , (28) here [arg R ± ( β )] C β means the change of the phase of R ± ( β ) as β goes along C β in acounterclockwise way. Let ℓ ± denote the loops on the complex plane drawn by R ± ( β ) when β goes along C β in a counterclockwise way. Then the values of w are determined by thenumber of times that ℓ ± surround the origin O . We note that w is not well defined wheneither ℓ + or ℓ − passes O , which means that the system is gapless.Now we demonstrate the bulk-edge correspondence for this winding number w in thenon-Hermitian SSH model with the matrix (20). With the values of the parameters as( t , γ , γ ) = (1 / , / , / t - t plane as shown inFig. 5(a). In this phase diagram, the white region represents a normal insulator (NI) with w = 0, and the blue region does a topological insulator (TI) phase with w = 1. For example,at the red dot in Fig. 5(a), from the trajectories ℓ ± as shown in Fig. 5(b), one can findthat the value of w is 1 since both ℓ + and ℓ − surround simultaneously the origin. Hencewe expect that the topological edge states appear in the TI phase. In fact, in the energylevels in a finite open chain with these parameters along the black arrow in Fig. 5(a), we canconfirm the appearance of the edge states (red in Fig. 5(d)) as expected. We note that thecontinuum energy bands in terms of the GBZ (Fig. 5(c)) agree with these energy levels exceptfor the topological edge states. In conclusion, we can establish the bulk-edge correspondencebetween the topological invariant defined by the GBZ and the existence of the topologicaledge sates in the non-Hermitian SSH model.
5. Topological semimetal phase with exceptional points (a) C β C β (b) Re( β )Im( β ) Re( β )Im( β ) Fig. 6 (a) Coalescence of the exceptional points. (b) Annihilation of the exceptional pointsat the cusp. The blue dots express the solutions of the gap-closing condition det H ( β ) = 0on the generalized Brillouin zone C β , meaning that these are the exceptional points. On theother hand, the yellow dots express those not on C β . The red stars are the cusps.The phase diagram in Fig. 5(a) also has a gapless phase (orange region). In fact, this gaplessphase appears because of its topological stability, inherent in 1D non-Hermitian systemswith the SLS and time-reversal symmetry (TRS) like the present model (17), and so it isa topological semimetal phase (TSM). Furthermore this phase appears as an intermediatephase between the NI phase and the TI phase characterized by the winding number (28).We note that for a real-space Hamiltonian (11), the TRS is defined as T H ∗ T − = H , where T is a unitary matrix satisfying T T ∗ = +1.We describe the reason why the SLS and TRS stabilize this TSM phase. Thanks to theSLS, the matrix H ( β ) can be written as an off-diagonal form (27), and the gap of thesystems closes at E = 0. We can obtain the condition for the gap closing as det H ( β ) = 0 rom the characteristic equation det [ H ( β ) − E ] = 0, and it is decomposed into two equationsdet R ± ( β ) = 0. These equations are polynomials of β and β − with real coefficients becauseof the TRS. Hence det R ± ( β ) = 0 can have solutions of complex-conjugate pairs, ( β, β ∗ ). Ifwe suppose β M and β M +1 form a pair of the complex-conjugate solutions of det R + ( β ) = 0(or det R − ( β ) = 0), they satisfy Eq. (16), meaning that E = 0 is in the continuum energybands, and the gap closes. Therefore the gap remains zero as long as this pair gives M th and( M + 1)th largest absolute values among the 2 M solutions of the equation det H ( β ) = 0. Inother words, as the values of system parameters changes, the GBZ is deformed so as to keepthe system gapless. Thus it is unique to non-Hermitian systems.According to the above discussion, the matrix (27) becomes the Jordan normal form atpoints on the GBZ where the gap closes. This means that these points are exceptional points,where some energy eigenvalues become degenerate and the corresponding eigenstates coa-lesce. Importantly, we can relate the motion of the exceptional points as shown in Figs. 6(a)and (b) to the change of the value of the winding number w . Namely, when the creation isby the inverse process of Fig. 6(a) and the annihilation is by the process of Fig. 6(b) (or viceversa), the systems undergo the topological phase transition from the NI phase with w = 0to the TI phase with w = 1 (or vice versa). We show the detail of this discussion in Ref. [32].
6. Summary
In summary, we reviewed the non-Bloch band theory in 1D non-Hermitian systems. Weexplain how to construct the GBZ, which is given by the trajectories of β M and β M +1 satisfying the condition | β M | = | β M +1 | for continuum energy bands, and show that the Blochwave number becomes complex in an infinite open chain in general. In addition, in non-Hermitian systems, the bulk-edge correspondence between the topological invariant andexistence of the topological edge states is established by using the GBZ.We can also show that in 1D non-Hermitian systems with the SLS and TRS, the TSMphase with exceptional points appear in terms of the non-Bloch band theory, and it isregarded as an intermediate phase between the NI and TI phases. Therefore the TSM phaseis stable, unlike Hermitian systems. Thus non-Hermiticity brings about qualitative changesto the topological phase transition.Finally we mention the experimental observation of the non-Hermitian skin effect in varioussystems. The previous work [27] experimentally realized a nonreciprocal tight-binding modelin a classical spring-mass system similarly to the simple model (1), and observed spatiallyasymmetric standing waves. After that, in Ref. [28], realizing the non-Hermitian SSH model(17) with γ = t = 0, the non-Hermitian skin effect was demonstrated by investigating thenon-unitary quantum walk dynamics. Furthermore the previous work [29] experimentallyalso realized the non-Hermitian SSH model (17) with γ = t = 0 by using an electric circuit.It showed the differences between the energy spectra in a periodic chain and those in anopen chain through an observation of the complex admittance. Acknowledgment
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